The report, from five Stanford graduate students and their faculty adviser, said the funds estimate average annual returns of 7.5% to 8%. The funds ought to use a more conservative calculation of 4.14%, the report said.

“You should not use an 8% rate when the liabilities are set in stone,” said Joe Nation, the faculty adviser in a telephone interview.

“Using that historical rate ignores the fan of outcomes,” said Howard Bornstein, one of the report's authors, in a telephone interview.

CalPERS disagreed with the methodology of the study.

“This study is an exercise in applying a new funny math to pension financing,” Pat Macht, a spokeswoman for the fund, said in an e-mail. “It ignores the reality of the last 20 years — that even in spite of several market downturns, CalPERS continued through a well-balanced, diversified portfolio to return 7.9% in investment earnings over the last 20 years.”

The report recommended that the state decrease benefits to retirees, increase future contributions from plan members and invest in less risky assets.

“This study reinforces the immediate need to address our staggering pension debt,” California Gov. Arnold Schwarzenegger said in a news release. “According to the study, California taxpayers are on the hook for over a half trillion dollars. That's nearly six times the size of our entire state budget.”

Mr. Schwarzenegger last year recommended creating a lower level of benefits for new hires, said Andrea McCarthy, a spokeswoman for the governor.